Critical Brownian multiplicative chaos

Abstract

Brownian multiplicative chaos measures, introduced in [Jeg20, AHS20, BBK94], are random Borel measures that can be formally defined by exponentiating $\gamma$ times the square root of the local times of planar Brownian motion. So far, only the subcritical measures where the parameter $\gamma$ is less than 2 were studied. This article considers the critical case where $\gamma =2$, using three different approximation procedures which all lead to the same universal measure. On the one hand, we exponentiate the square root of the local times of small circles and show convergence in the Seneta--Heyde normalisation as well as in the derivative martingale normalisation. On the other hand, we construct the critical measure as a limit of subcritical measures. This is the first example of a non-Gaussian critical multiplicative chaos. We are inspired by methods coming from critical Gaussian multiplicative chaos, but there are essential differences, the main one being the lack of Gaussianity which prevents the use of Kahane's inequality and hence a priori controls. Instead, a continuity lemma is proved which makes it possible to use tools from stochastic calculus as an effective substitute.

arXiv commentPublished version. 49 pages

References in corpus (9)

• Convergence in law of the minimum of a branching random walk
• Critical Gaussian multiplicative chaos: Convergence of the derivative martingale
• Extreme local extrema of two-dimensional discrete Gaussian free field
• Convergence of the centered maximum of log-correlated Gaussian fields
• Critical Liouville measure as a limit of subcritical measures
• Subcritical multiplicative chaos for regularized counting statistics from random matrix theory
• Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation
• Planar Brownian motion and Gaussian multiplicative chaos
• Thick points of random walk and the Gaussian free field

Cited by in corpus (6)

• Multiplicative chaos of the Brownian loop soup
• Thick points of 4D critical branching Brownian motion
• Crossing exponent in the Brownian loop soup
• A limit law for the most favorite point of simple random walk on a regular tree
• Favorite sites for simple random walk in two and more dimensions
• Convergence for Complex Gaussian Multiplicative Chaos on phase boundaries